Maximum odd induced subgraph of a graph concerning its chromatic number

Let fo(G) ${f}_{o}(G)$ be the maximum order of an odd induced subgraph of G $G$. In 1992, Scott proposed a conjecture that fo(G)>= n chi(G) ${f}_{o}(G)\ge \frac{n}{\chi (G)}$ for a graph G $G$ of order n $n$ without isolated vertices, where chi(G) $\chi (G)$ is the chromatic number of G $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that fo(G)>= 2n4 ${f}_{o}(G)\ge 2\unicode{x0230A}\frac{n}{4}\unicode{x0230B}$ for a connected graph G $G$ of order n $n$. Scott's conjecture is open for graphs with chromatic number at least 3.